A Linear Matrix Inequality (LMI)

where

and

and

is given.

is a vector in

.

...and thus is a vector in . What does it mean that a vector is greater/less than zero? Tonio
Then, the linear map

from

to

, where

is a space of symmetric matrices with prescribed block diagonal structure, is associated such that

. The range of this map is

.

My Question is, how to obtain

??

I guess, that

is a matrix which columns are the matrices

.

And

is then the column space of

??

And when I want to find a matrix which is orthogonal to

, is it then a matrix which is orthogonal to

??

As you might have noticed, I've no idea what to do.

So, please help me.

thanks